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Your data matches 68 different statistics following compositions of up to 3 maps.
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Matching statistic: St000740
(load all 148 compositions to match this statistic)
(load all 148 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 3
[2,3,1] => [3,1,2] => 2
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => 4
[1,3,4,2] => [1,4,2,3] => 3
[1,4,2,3] => [1,3,4,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 4
[2,1,4,3] => [2,1,4,3] => 3
[2,3,1,4] => [3,1,2,4] => 4
[2,3,4,1] => [4,1,2,3] => 3
[2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [4,1,3,2] => 2
[3,1,2,4] => [2,3,1,4] => 4
[3,1,4,2] => [2,4,1,3] => 3
[3,2,1,4] => [3,2,1,4] => 4
[3,2,4,1] => [4,2,1,3] => 3
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [4,3,1,2] => 2
[4,1,2,3] => [2,3,4,1] => 1
[4,1,3,2] => [2,4,3,1] => 1
[4,2,1,3] => [3,2,4,1] => 1
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [3,4,2,1] => 1
[4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 5
[1,2,4,5,3] => [1,2,5,3,4] => 4
[1,2,5,3,4] => [1,2,4,5,3] => 3
[1,2,5,4,3] => [1,2,5,4,3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => 5
[1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,4,2,3,5] => 5
[1,3,4,5,2] => [1,5,2,3,4] => 4
[1,3,5,2,4] => [1,4,2,5,3] => 3
[1,3,5,4,2] => [1,5,2,4,3] => 3
[1,4,2,3,5] => [1,3,4,2,5] => 5
[1,4,2,5,3] => [1,3,5,2,4] => 4
[1,4,3,2,5] => [1,4,3,2,5] => 5
[1,4,3,5,2] => [1,5,3,2,4] => 4
[1,4,5,2,3] => [1,4,5,2,3] => 3
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001291
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 4
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000054
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 2
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 3
[1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3] => [2,1,3] => [3,1,2] => 3
[2,3,1] => [3,1,2] => [2,1,3] => 2
[3,1,2] => [3,2,1] => [1,2,3] => 1
[3,2,1] => [2,3,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 4
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 3
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 4
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 3
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 4
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 3
[2,4,1,3] => [4,3,1,2] => [2,1,3,4] => 2
[2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 2
[3,1,2,4] => [3,2,1,4] => [4,1,2,3] => 4
[3,1,4,2] => [4,2,1,3] => [3,1,2,4] => 3
[3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 4
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 3
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,4,2,1] => [4,1,3,2] => [2,3,1,4] => 2
[4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 1
[4,1,3,2] => [3,4,2,1] => [1,2,4,3] => 1
[4,2,1,3] => [2,4,3,1] => [1,3,4,2] => 1
[4,2,3,1] => [2,3,4,1] => [1,4,3,2] => 1
[4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 5
[1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => 4
[1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 3
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 5
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 4
[1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => 5
[1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => 4
[1,3,5,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => 3
[1,3,5,4,2] => [1,4,5,2,3] => [3,2,5,4,1] => 3
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 5
[1,4,2,5,3] => [1,5,3,2,4] => [4,2,3,5,1] => 4
[1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 5
[1,4,3,5,2] => [1,3,5,2,4] => [4,2,5,3,1] => 4
[1,4,5,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 3
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St001184
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [2,1] => [1,1,0,0]
=> 2
[2,1] => [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 3
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000051
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> 0 = 1 - 1
[1,2] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
[2,1] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[1,2,3] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[1,3,2] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,1,3] => [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[2,3,1] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[1,3,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[1,4,2,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[1,4,3,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[2,1,3,4] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
[2,1,4,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[2,3,1,4] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> 3 = 4 - 1
[2,3,4,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[2,4,1,3] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[2,4,3,1] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[3,1,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
[3,1,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[3,2,1,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 3 = 4 - 1
[3,2,4,1] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[3,4,1,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[3,4,2,1] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[4,1,2,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[1,2,4,3,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> 4 = 5 - 1
[1,2,4,5,3] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[1,2,5,3,4] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
[1,2,5,4,3] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
[1,3,2,4,5] => [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> 4 = 5 - 1
[1,3,2,5,4] => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> 3 = 4 - 1
[1,3,4,2,5] => [2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> 4 = 5 - 1
[1,3,4,5,2] => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[1,3,5,2,4] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
[1,3,5,4,2] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
[1,4,2,3,5] => [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> 4 = 5 - 1
[1,4,2,5,3] => [2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> 3 = 4 - 1
[1,4,3,2,5] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 4 = 5 - 1
[1,4,3,5,2] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> 3 = 4 - 1
[1,4,5,2,3] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
Description
The size of the left subtree of a binary tree.
Matching statistic: St000141
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3 = 4 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000147
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[2,1] => [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4 = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 5 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 5 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 3 - 1
Description
The largest part of an integer partition.
Matching statistic: St000316
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3 = 4 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St001227
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 1
[1,2] => [2,1] => [1,1,0,0]
=> [2,1] => 2
[2,1] => [1,2] => [1,0,1,0]
=> [1,2] => 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 4
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 3
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 4
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 3
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 4
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 4
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 3
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001497The position of the largest weak excedence of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000010The length of the partition. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000067The inversion number of the alternating sign matrix. St000133The "bounce" of a permutation. St000204The number of internal nodes of a binary tree. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000444The length of the maximal rise of a Dyck path. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000989The number of final rises of a permutation. St000676The number of odd rises of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000005The bounce statistic of a Dyck path. St001927Sparre Andersen's number of positives of a signed permutation.
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